Is x^2 - y^2 a positive number?
(1) x - y is a positive number.
(2) x + y is a positive number.
Answer is C
(1) x - y is a positive number.
(2) x + y is a positive number.
Answer is C
Solution
x^2 - y^2 = (x+y)*(x-y).
Obviously if x^2 - y^2 is positive, either both (x+y) and (x-y) are positive or both are negative.
Consider first (1) alone.
We know (x-y) is positive but we do not know anything about (x+y).
For example consider x= 2 and y= -3. So (x-y) = 2 + 3 = 5 (a positive number) and (x+y)= -1.
So x^2 - y^2 = (x+y)*(x-y) = -5 which is a negative number.
If x= 7 and y = 5, then (x-y) = 2 (a positive number)and (x+y) = 12.
Also x^2 - y^2 =(x+y)*(x-y) = 12 * 2 = 24 which is a positive number.
So nothing definite can be said.
Hence (1) is not enough.
Consider (2) alone.
We know (x+y) is positive, but we know nothing about (x-y).
For example let x = 8 and y = 5, then (x+y) = 13(a positive number) and (x-y) = 3.
So x^2 - y^2 = (x+y)(x-y) = 39 (a positive number).
If x = 5 and y = 8 then (x+y) = 13 a positive number and (x-y) = -3
So x^2 - y^2 = (x+y)(x-y) = -39 (a negative number).
Since nothing definite can be said (2) alone is not sufficient.
On combining we have that both (x+y) and (x-y) are positive and so their product (x^2 - y^2) will also be positive.
So the answer to the question is yes.
The correct answer is (C).
